It is possible, at least in Riemannian manifolds with the Levi-Civita connection (I'm not sure if this is more general), to find local coordinate systems for every point $P$, such that the coordinates curves are geodesics. This is done thank to the exponential map for Riemannian manifolds. In general relativity theory it corresponds to choosing a local inertial frame (the observer moves along a geodesic...).
In this coordinate system, called Riemann normal coordinates, the Christoffel symbols are all null, and the Riemannian metric tensor corresponds to the identity matrix (keep an eye: only in $P$).
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Author of the notes: Antonio J. Pan-Collantes
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